The higher-order and multi-lump waves for a (3+1)-dimensional generalized variable-coefficient shallow water wave equation in a fluid

Hu, Cong-Cong; Tian, Bo; Qu, Qi-Xing; Yang, Dan-Yu

doi:10.1016/j.cjph.2021.10.022

Cited by 10 publications

(5 citation statements)

References 49 publications

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“…For instance, Deng et al studied the soliton solutions, integrability and periodic of a generalized (3+1)-dimensional variablecoefficient nonlinear-wave equation [33]. The lump solutions of nonlinear evolution equations have always been a key topic of research in the field of mathematical physics due to its wide applications in optical media [34,35], plasma [36,37], Bose-Einstein condensates [38] and water waves [39][40][41]. For this reason, many methods have been proposed to find the lump solutions, such as Hirota bilinear method [42], long wave limit method [43], quadratic function method [44], etc.…”

Section: Introductionmentioning

confidence: 99%

Three-wave lump solutions and their dynamic behaviors for the (3+1)-dimensional constant-coefficient and variable-coeffcient differential equations

Feng,

Zhao

2024

Phys. Scr.

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In this paper, we propose two theorems to illustrate the types of equations that can be solved using the quadratic function method to derive the lump solutions localized in the whole plane, which are called three-wave lump solutions, and provide two constant-coefficient equations to illustrate. We further extend the quadratic function method to the variable-coefficient differential equations and obtain the three-wave lump solutions for two (3+1)-dimensional variable-coefficient equations. Moreover, the amplitudes of these lump waves and the distances between the two valleys of each lump are also obtained. Meanwhile, the motion trails, displacements and the velocities of these lump waves are analyzed in detail by virtue of numerical simulation. The study can be used to describe the motion of nonlinear waves in shallow water under the influence of time, and the results can enrich the types of solutions for the KdV-type equations. In addition, the 3d plots and corresponding density plots of the lump waves are displayed to show their spatial structures.

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Section: Introductionmentioning

confidence: 99%

Three-wave lump solutions and their dynamic behaviors for the (3+1)-dimensional constant-coefficient and variable-coeffcient differential equations

Feng,

Zhao

2024

Phys. Scr.

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“…They are also known as giant waves, whose wave heights are often several times the wavelengths, and are localized in time and space [11]. Rogue waves can explain interesting nonlinear phenomena in different physical contexts via rational solutions of partial differential equations, such as Fokas equation [12], Gardner equation [13], shallow water wave equation [14] and so on [15,16]. Lump solutions are localized waves, decaying algebraically to an asymptotic value and moving with unvarying velocity [17].…”

Section: Introductionmentioning

confidence: 99%

On the study of the higher-order and multiple lump/rogue waves to the (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation

et al. 2023

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In the text, rational solutions to a (3+1)-dimensional variable-coeﬃcient Kadomtsev-Petviashvili equation in determinant form are solved by means of the Kadomtsev-Petviashvili hierarchy reduction method and Hirota’s bilinear form. The ﬁrst-order, higher-order, multi-lump waves and multi-rogue waves on the basis of the solution in Gramian are presented in graphs. Periodic and parabolic line rogue waves are obtained by choosing diﬀerent dispersion coeﬃcient functions. The second-order rogue waves are also depicted, which can be seen as the composition of two ﬁrst-order rogue waves. For multiple rogue waves, taking the periodic and parabolic line rogue waves as examples, it can be observed the change tendency of amplitudes of the intersection region is diﬀerent from that in branches of line rogue waves. Besides, propagation of the ﬁrst-order lump wave, interaction between one-lump-soliton and one-rogue-wave, and two kinds of collisions of two lump waves, i.e., elastic and inelastic, are also shown in the x - t plane.

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“…Considering the introduction of variable coefficients into equation (1), researchers can study some phenomena in the inhomogeneity of media and the boundaries [10][11][12][13][14][15][16]. For example, recent studies have systematically investigated the soliton solutions of perturbed Fokas-Lenells equation with variable coefficients [17].…”

Section: Introductionmentioning

confidence: 99%

“…For example, recent studies have systematically investigated the soliton solutions of perturbed Fokas-Lenells equation with variable coefficients [17]. In this paper, we will focus on a variable-coefficient version [10][11][12][13][14][15][16] of equation (1)…”

Section: Introductionmentioning

confidence: 99%

“…The lump solutions for equation (1) have been obtained [14]. Higher-order, multi-lump waves [15], and soliton [16] for equation (1) have also been given by KP hierarchy reduction method. When α 1 (t) = 2, α 2 (t) = 1, α 3 (t) = α 4 (t) = 3, equation (1) becomes the (3+1)-dimensional Jimbo-Miwa equation, which represents some (3+1)-dimensional physical phenomena [2].…”

Section: Introductionmentioning

confidence: 99%

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Dynamic mechanism of nonlinear waves for the (3+1)-dimensional generalized variable-coefficient shallow water wave equation

Sun

2022

Phys. Scr.

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Under investigation in this paper is a (3+1)-dimensional generalized variable-coefficient shallow water wave equation, which can be used to describe the flow below a pressure surface in oceanography and atmospheric science. Employing the Kadomtsev-Petviashvili hierarchy reduction, we obtain the breather and lump solutions in terms of Grammian. We investigate the generation mechanism and conversion of the breathers, lumps and rogue waves. We find that the breather is produced by the superposition of three parts: The soliton part, the periodic wave part and the background part. The angle between the soliton part and the periodic wave part affects the shape of the breather. Considering the influences of the variable coefficients, we observe the breathers propagating on the periodic backgrounds, with double peaks and the breathers propagating periodic with time, respectively. Taking the long-wave limits, we get the rational solutions which describe the lumps. We find that the characteristic lines keep unchanged on the x − y plane, which means that the lump is similar to a part of the breather. Linear rogue waves only appear on the y − z plane.

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The higher-order and multi-lump waves for a (3+1)-dimensional generalized variable-coefficient shallow water wave equation in a fluid

Cited by 10 publications

References 49 publications

Three-wave lump solutions and their dynamic behaviors for the (3+1)-dimensional constant-coefficient and variable-coeffcient differential equations

Three-wave lump solutions and their dynamic behaviors for the (3+1)-dimensional constant-coefficient and variable-coeffcient differential equations

On the study of the higher-order and multiple lump/rogue waves to the (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation

Dynamic mechanism of nonlinear waves for the (3+1)-dimensional generalized variable-coefficient shallow water wave equation

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